Switched Mode Power Conversion Prof. L. Umanand Department of Electronics System Engineering Indian Institute of Science, Bangalore Lecture - 39 Magnetic Design Good day to all of you. Today, we shall discuss on one more important aspect of DC-DC converter, which is the magnetic components. Recall in the last class that we mentioned that to design a DC-DC converter, three main components and the selection are important. One being the power semiconductor switch, the other one is the capacitor and third is the magnetic component, both the inductor and the transformer in case of isolated DC-DC converters. So, today the objective is that we try to understand something about the magnetics, how we would go about designing this inductors and transformers such that it can be included in a practical DC-DC converter. (Refer Slide Time: 01:32) So, the magnetics have always been a domain, which is, which has been a pretty µch difficult to understand and implement. However, once a design is performed with magnetic components, the magnetic component being passive, they are very, very robust. Nothing happens to them unlike the power semiconductor switches or the capacitors. Both of them which may blow once the rating is exceeded. In a magnetic component, the only rating
that can be exceeded is the currents and the saturation levels which will cause other components to get damaged, not itself. So, let us see the magnetic components from two viewpoints. One view point is the electrical and the other is the magnetic domain. So, these two are tightly interfaced and they interact through a window through which there is energy transaction. So, let s say we call this as the energy port or the power port. So, this is the port. It is like all ports, electrical ports. It has 2 conductor points, 2 wires and 2 variables. So, on the electrical side, we have v, the voltage and i, the current variables. Now, v is the potential across the port and i is the current through the leads of the port wire. And now on the other side, you have a potential variable and a flow variable. The potential variable is called mmf, the magneto motive force. The flow variable is called d dt. (Refer Slide Time: 05:34) Now, these domains are linked together by two very powerful and fundamental laws, the Faraday s equation. You also have the ampere s rule or equation. Now, these two important rules, principles govern the entire interaction between the electric domain parameters and the magnetic domain parameters. So, let us say we have the potential variable v voltage across the port and i is the current through the port. On one side, you have the mmf magnetic motive force, the potential variable on the magnetic domain and
rate of change of flux, not flux remember, rate of change of flux. In fact, mmf is flux by reluctance. Sorry, Mmf is flux into reluctance and the flow is the rate of change of flux. So now, how are these variables related? The relationship, the linking is between the cross variables. The potential variable on the electrical side is linked to the flow or the kinetic variable on the magnetic side. The kinetic variable on the electrical side is linked with the potential variable on the magnetic side. Both these linking happen, take place with the parameter N. What is this N? N is nothing but the number of turns of the winding. So, to make sense out of what we have discussed till now, let us see how practical magnetic components look like and try to understand what it is. Now, here I have a piece of magnetic component. Observe this. (Refer Slide Time: 09:01) This is the core. This is a material, ferrite material. And here, in between, you see the bobbin. On the bobbin, you see the windings. These are enameled copper windings. You can remove the core. The core is now split into two halves, the bobbin along with the copper windings and the removed copper winding is having 2 terminals here. Now, we have two halves of the core. Now, this copper, current carrying copper conductor which has N turns and this N turns is wound on to, is wound around the magnetic core material.okay. So, let us get back to this later on, but from now assume that we are able
to wind the copper coils as shown. That is the number of turns, N that I have been talking about. (Refer Slide Time: 10:50) So, now, coming back to the white board; so N is number of turns, copper turns which are wound on the core, magnetic core. And how are these related? So, V = N d. This is called Faraday s law. The voltage across the winding is equal to N times the rate of change of flux within the core. That is the first. Now, to link the mmf and the current, we have the Ampere s rule which states Ni, which is, which is the ampere turns are equal to mmf. So, this will come as the, as the result of the Ampere s rule. So, the ampere law, by Ampere s law, we have i, which is H. dl, where l is the magnetic length. Integrating over the whole length, you would have i, which is H. dl m. Now, if there are many turns, the force within the core is a cuµlative effect of all the turns. Therefore, N comes into the picture. So, it is Ni cuµlative effect of all the turns, which is the H. dl m. Therefore, Ni which is equal to Hl m and the force field with in the magnetic field within the magnetic core is given by N i /l m. So, this is as per the Ampere s rule. From the Ampere s rule, we have these. From faraday s law, we have these. Now, these 2 laws are the most critical principles on which the entire magnetic field and the electrical domain interaction rests. dt
(Refer Slide Time: 14:31) So, now, V = N d. Mmf is equal to Ni. These 2 rules are the foundations of which all dt other magnetic based equations that are used in design. Now, take example. The power, power is vi. And in the magnetic domain, what is it? In the electrical domain, you have a potential variable, µltiplied with the kinetic variable, which is a power, watts. Now, let us take Faraday s equation. Substitute it for the potential variable v. So, you have(n d dt ) iis the power in watts. Now, from the second equation, mmf is equal to Ni. Let us combine these two and you will get d mmf is again power in watts. You see that the power in watts in magnetic dt domain is also composed of the 2 variables, the potential variable mmf and the kinetic variable or the flow variable d, product of which is watts power. dt
(Refer Slide Time: 17:12) Now, in the literature, you would see a very important graph B versus H. B is called flux density, which is flux through the core by core cross section area. And H, we saw coming from Ampere s rule N i /l m. Now, this would normally be a straight line like that. But, it is not straight through out. After some stage, the flux saturates. There can be no d. Not only does it saturate the core, magnetic core has memory; it demonstrates a clear cut hysteresis curve in this manner. However to understand, we shall consider the virgin specimen without the hysteresis, so that you get a clear understanding of what is happening within the core. So, let us take out the hysteresis portion. Now, this slope B / H is called µ, the permeability. B which is equal to µh, B is nothing but flux by the area cross section of the core µ N i /l m l m. So, flux which is equal to N i /(l m /μ A). This is nothing but Ni by reluctance or mmf by reluctance. This is one very important relationship. dt
(Refer Slide Time: 20:29) Flux is equal to mmf by reluctance. In the literature, many a times, it is called ohm s law of the magnetic circuit. However, remember that in the ohm s law, this R is not the R, the dissipative resistance. It is more related to Q = CV. So, this is more related to Q = forµla, rather than I = V / R. Now, this is the electrical equivalent. You can put it in that form one more term called permeance, which is 1 mmf. Now, what is 1? I am going to introduce Reluctance Reluctance 1 Reluctance So, permeance is more closer to your equivalent capacitance concept. So, = mmf. Now, compare these 2. Flux is like the charge, mmf-the potential, permeance is like the capacitance. So, the reluctance is 1 / capacitance; 1 / C equivalently.
(Refer Slide Time: 21:19) So, now, let us go further. So, flux is permeance into mmf, which is Ni. So, N d that is µltiplying both sides by N, N 2 di ; and differentiating with respect to time. So, what do you dt see? You see that this from Faraday s law is voltage v and this is equal to something di dt, which is equal to inductance L di (Refer Slide Time: 24:32) dt. So, we define N2 as inductance value. dt So, inductance L is N 2. This is the important relationship because in many of the core catalogs, this permeance value is given as what is called as AL factor. It is has the unit of
nano Henry per turns square. So, this is given in the datasheets in many of the gaped cores or the inductor type cores. This permeance is used to design a number of turns. Now, L value is obtained from your DC-DC converter electrical design, based on the amount of current ripple that you want to have, the voltage across the inductor and all those things. The L value is obtained from electrical design. Then, N, the number of turns you need to put for the particular core would be L / λ permeance square, square root, because this is nano Henry per turns square. Root of that will give you the number of turns of wind that you need to get the value of N. So, this is a very important relationship. We can go one step further. (Refer Slide Time: 26:35) L is equal to N 2, which can be written as N 2 Reluctance. 1 Reluctance = μa c l m., mean magnetic length N 2. µ is nothing but µ0 and relative permeability, core cross section area, N 2 by the magnetic length. This is another useful relationship; in case you are unable to find the value of permeance from the core catalog or in case you are using a transformer core for which do not know the permeance. You are using a transformer core to build an inductance in which you need an air gap; in such cases, this relationship is useful. This is basically derived from the same relationship, only that you know the play of variables within the play of core cross section area and magnetic length core, which is given. Relative permeability for core material is known. This is the permeability for free
space, which is 4π 10-7 Henry per meter. So, this way also inductance L, the inductance L can be found out, but as the inductance L is obtained from electrical side, L is known, core cross section is known from the datasheets of the core material, µ r permeability, relative permeability is known from the core material data, mean magnetic length is known. From that, you can find out N. (Refer Slide Time: 29:05) So, let us see how we now go about finding the inductor to the physical parameter in a logical way. So, on the mechanical side, we have what is known as core cross section area. You have the window area, you have B sat, again coming from the material. And on the electrical side, we know how to get the value of the inductance. We know what is the current going through the inductance. We know what the voltage is across the inductance. From here we know what the energy in joules ; which the inductor will have to store. Now, from the electrical parameters, we need to arrive at the physical parameters such that you can select a physical core for winding the material. Apart from this, we also need to know what the number of turns is, what is the gauge of the copper wire? That is the thickness of the wire and the diameter of the wire, which we need to wind. Now, these physical parameters, we have to relate to the electrical parameter.
(Refer Slide Time: 31:06) So, first let us say from the electrical side, we have the energy, which can be easily found out. The energy, the energy E = 1 2 LI m 2. This is the maximum energy that the inductor will have to store. What is this Im? Typically, in inductor, DC-DC convertor, if you plot the inductor current, it is of this form. So, the current rises during dt and falls during 1 - d t. This is Δ il. Let us say that this is most of the times I0, if it is a buck converter. If it is a boost converter, it is I in. So, let us say that this is the average value. So, where is the maximum energy? When the current is maximum, you will see that the maximum energy is stored. So, this value is nothing but I avg + i l plus this. So, Im is I 2 avg + i l and use that here to calculate 2 the energy. Now, all this is calculation in the electrical side. Let us try to relate this to the physical parameters. (Refer Slide Time: 33:40) Let us makes a split, E = 1 LI 2 m 2. is 1 LI 2 m I m.. Let us make a split. Let us take this from 2 different, from the 2 different rules that is one is the flow based and another is the potential based. If you take the Faraday s equation, where we know that
V = N d di, which is equal to L. dt dt As the current is linear, you can as well write it as N m = LI m. Because, in the same time, a given time, if the current has reached Im, the flux would have reached m. N m is written as NAcBm. B m is the flux density that is equal to LI m. So, this we can incorporate here. Let us come to the other aspect. What to do for this? Let us take, let us take another physical parameter. Notice, here we are trying to relate to this physical parameter that is core cross sectional area. Let us try to relate to this the other one, the window area. Now, let us see what the window area is. (Refer Slide Time: 36:11) If you see, we have this core. Let we take simple EE core which is like this. So, this is EE core. You have two EE joined together. And, the former comes in here. The red line indicates the former. The red line indicates the former. Also, this is, on the former is wound the windings. So, you will see windings wound in this fashion. And you will see the other end of the windings also here. So, this is how it would appear here. To picture this better, let us go back again and look at the inductor core. So, you look at this core. Let me take out the bobbin. I will keep it aside. Then, we have this core. Now, this is what I have been drawing just now; two halves which are joined together. And you see here this is one opening and this is another opening.
The wire passes through, goes over and comes out through the bottom. How is the former introduced? You take this former and you are introducing it like this. So, you see here the white portion is the former portion and on the former is wound the copper wire. So, if I take a section cut, if I take a section cut, you would see exactly the figure I have drawn there with many windings. Now, this winding has to fit into this area. Let me put this together. The entire winding will have to fit into this area and that is called the window area. This is called the window area because other portion open here, other portion opening here is nothing but carrying the other half of the turn. Therefore, in one complete turn, half of the turn is sitting in this opening and other half is sitting in this opening. So, you consider only you one opening as something available to you. So, this window area is called Aw. Coming back to the white board. So, this window area is called Aw. Now, within this Aw window area, what should fit? All these N turns should fit. Now, each of these entire N terms have a cross section area. Now, let us say, it has across section area a wire. a wire is decided based on this thickness of the wire. The thickness of the wire is decided based on amount of current that a wire has to carry. The amount of current of course is obtained from the electrical circuit side by electrical analysis. So, a wire is the thickness of the; is the cross section area of the wire. N of them should fit into this window area. So, N of them should fit into this area. So, Aw should be greater Na wire. Now, let us some simple math.
(Refer Slide Time: 42:55) Aw should be greater than Na wire. Normally, the amount of current that is passed in, in through the copper conductor is based on experience. So, let us say these are all current lines through. So, the current i which flows through, now there is RMS component of current, which causes heat to be dissipated in the copper conductor. If the current is very dense, meaning for a given area, cross section area, the RMS current is very high, then conductor can be very hot and lead to lot of i 2 R losses. Therefore, there is an appropriate amount of current density that is has to be allowed. So, let us call J as the current density. It is usually expressed terms of amps per meter square or amp per mm square. So, normally, an amount of 3 amp per mm square or 3 10 6 amp per meter square is the amount of current density that is consider as nominal.
(Refer Slide Time: 46:46) So, which means that when you say 3 amps per mm square for every 1 mm square of this cross section area, 3 amps is allowed to flow through that cross section area to keep that wire reasonably cool. However, you will find designers taking a conservative value, 2 amps per mm square for very high powers to up to 5 amps per mm square for very low powers. This will lead to copper being cooler. This will lead to copper being hotter. You have more I 2 R losses with higher current density. So, J the current density is I rms by the area of cross section of the wire. Now, I rms can also be written in terms of a crest factor. I rms is nothing but Im, peak value by a crest factor by a wire, just put everything in terms of peak value. Therefore, the wire cross section area can be calculated using, using this above equation, I rms / J. Now, we are coming back to this issue of J and this, let us put in this fashion. Im, that is I am referring to this equation equals J K c a wire. Now, this is this is another interesting relation that may be used. Now, going back to this, we wanted this inequality to be satisfied. What is that inequality?
(Refer Slide Time: 49:49) It says Aw, the window area should be greater than the number of turns times a wire. And from the discussion just concluded, we could say N a wire, is nothing but I rms / J. It can be written as NIm by crest factor J. So, let us find Im as something that is should be equal to Aw Kc J / N. Let us here, there is one more term that we need to add. Keep this in back of the mind; one more a practical term. (Refer Slide Time: 50:29) You saw that when you take the wire, the bobbin, when you start winding, you have the wires cross section, which appears like this. I will consider just only the one portion. Other
portion is a duplicate, other half of the winding. So, let us say, you put the layers of the copper winding in this fashion. There are many gaps here because the windings are circular cross section. And then, you will see a lot of air gaps. And there is also, the thickness of the bobbin itself, which is eating into the window area space. The thickness of the bobbin, which eats into the window area space, the air gaps which eat into the window area space and this wire itself; if you take this wire, if I blow this up, this cross section is having two parts. One is the outer sheath, which is the enamel because the windings are sitting next to next. You do not need, you do not need short circuits happening between the windings. Therefore, the windings are, the wires are covered with coating of enamel, which provides a natural insulation. Therefore, the wire cross section, what we say as a wire, is π r 2, where this r is actually less than the total wire gauge radius because of this enamel which eats away some area. This is the thickness of enamel. Now the air gaps, the bobbin itself, the enamel of the wires all eat into, eat up into the window area plus there is also an additional issue which is the skill of the winding. Now that also eats up into the window area. Therefore, what is available is not Aw. What is available is just KwAw, where Kw is a factor which is less than 1. So, this factor is arrived at empirically. It is around 0.6 for inductors, properly wound inductors. And for transformers, it would be anywhere from 0.2 to 0.4. It would be lesser because when you talk of transformers, tt is µltiple windings. So, there will be a layer of insulation, which is also eats up into the same window area. So, what is actually available is KwAw.
(Refer Slide Time: 54:46) So, therefore, if I rewrite that, KwAw should be greater than N a wire, which is the copper cross section, wire cross section. So, which is basically, if I use the equality NIm by crest factor J or Im can be written as Kw Kc J Aw / N. So, here we see that we are relating it with the other window, other area, the window area. (Refer Slide Time: 55:45) Now, coming back to our energy equation 1 2 L mi m I m, which is half. Now, this was
NAc B m (K ck wa w J / N). So retaining these two physical parameters on one side, you will get Aw A c, which is called area product as two times energy value / Kc Kw J Bm. This is an important equation, energy obtained from the electrical circuit parameters, Bm is from the core material, which is if it s a ferrite core it is 0.3 at saturation. And the operating flux density may be around 0.25, J- 3 ampere meters square; Kw of around 0.6, Kc is 1 in this case, and you can calculate the area product. Once you have the area product from the catalog of the cores - core catalog you run through the area product value and pick that area product value, which is just more than calculated. That would be the core that will be used for the inductance. So, at this point, we know how to select the core for the inductance. In the next class, we will look at how we introduce air gaps, if necessary and choice of the wire gauges and then see how we modify the design for a transformer too. Thank you.